Use of -거나 in this question

Question

This is question 1 from the 35th Topik 2 (제 35 한국어능력시험)

1. 어제는 친구들과 같이 점심을 ( )바로 도서관에 갔다.

1 먹든지                                          2 먹거나

3 먹고 나서                                     4 먹다 보면

The answer key says that the right answer is 2. 먹거나.

However, looking at the entry of -가나 in a mobile dictionary app shows the following sample sentence:

그것은 있거나 없거나 마찬가지다.
(It doesn't matter whether I have it or not)

Other provided examples are similar. The Naver online dictionary entry also gives a similar definition.

However, it doesn't look like -거나 has been used in the same sense in the above question.

What is the meaning of -거나 and how has it been used here?

What exactly does the sentence in the question mean? Does it mean he went to the library as soon as he finished lunch? Can the above sentence be written as shown below?

1. 어제는 친구들과 같이 점심을 먹는 후 바로 도서관에 갔다.
2. 어제는 친구들과 같이 점심을 먹자마자 후 바로 도서관에 갔다.

Answer 1

As jick said, the answer key is wrong. The correct one is 3.

Bear in mind that -거나 means "or", not "then" or "and".

I see that you posted the link from Naver. Just want to remind you not to mix up 거나 and -거나 ending which is to be added after verbs.

거나 is short for 것이나, namely formed by 것 + (이)나 (particle also meaning "or", but added after noun), while -거나 is directly added after verbs/adjectives.

Plus, the two sentences you provided at the last part can be accepted.

FYI, here is another answer key which shows the answer is 3.

https://drive.google.com/file/d/0B8EMStKesfgtSlB3YkpvX1BtbVk/view

Answer 2

어제는 친구들과 같이 점심을 ( )바로 도서관에 갔다.
1 먹든지
2 먹거나

3 먹고 나서
4 먹다 보면

The right answer is 3. 먹고 나서